Spectroscopy: Photoemission and other techniques - Ondrej pr

Spectroscopy:

Photoemission and other techniques

Ondrej Sipr

VI.

NEVF 514 Surface PhysicsWinter Term 2011 - 2012

Troja, 18th November 2011

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

How to study microscopic (nanoscopic. . . ) structure

Dimensions of the probe have to be comparable to the dimensionsof the investigated system.

One cannot see inside the material: wave-length of visible light isby orders of magnitude larger than interatomic distances.

Only indirect methods are available if you want to study interiourof a material: blackbox techniques.

BlackboxOutputInput

Diffraction: studying geometric structure (positions of ions)

Spectroscopy: studying electronic structure (energies εnk andcrystal momenta k of electrons)

Modeling and comparing with theory is often indispensable.

Density of states

Recall:

Density of states (DOS) describes how many electron states arethere at certain energy ε.

n(ε) =∑

n

∫

1BZ

dk

4πδ(ε − εnk) .

Knowing the DOS for a solid is analogous to knowing the energylevels for an atom.

If we want to know other quantum numbers of stationary states(e.g., crystal momentum k), we need to study more than just DOS.

Photoemission as a probe of occupied DOS

EF

Exciting electrons from the valenceband.

Higher DOS for particular energy E

means that more electrons with thatenergy are available for beingexcited.

More electrons at our disposalmeans higher intensity of thephotoemission spectrum.

Emission and absorption of light quanta

◮ Assume a system is described by an unperturbed HamiltonianH0. Consider its two eigenstates |ψi 〉 and |ψf 〉 (i.e., stationarystates).

◮ This system is perturbed by Hamiltonian Hint which representsthe interaction between the electrons and the photons,

Hint = −e

mA(x) · p .

The vector potential A(x) represents the quantizedelectromagnetic field and p is the momentum operator actingon the electron states.

The above Hamiltonian makes used of the so-called Coulomb

calibration (not important for us). Moreover, it neglects terms

proportional to |A|2, which may be important — we cannot describe

two-photon transitions in this way.

Fermi’s Golden Rule

Probability (per unit time) that a perturbation Hint causes atransition between states |ψi 〉 and |ψf 〉 of the unperturbedHamiltonian is, within the first order perturbation theory, given by

w =2π

~|Mfi |

2 δ(Ef − Ei ± ~ω)

(“+” for absorption of photons, “-” for emission of photons).

Transition matrix element is proportional to

Mfi ≈⟨

ψf

∣

∣

∣e± i

~q·x ǫ · p

∣

∣

∣ψi

⟩

,

where q is the photon wave vector (cq = ~ω), ǫ is the polarizationvector of the radiation.

How the DOS enters the business

Probability of a radiative transition

w =2π

~|Mfi |

2 δ(Ef − Ei ± ~ω) .

Recall: n(ε) =∑

n

∫

(1BZ)dk4π δ(ε− εnk).

Photoemission: we are interested in probability that a final state“f” is created via incoming radiation. Therefore, we have tointegrate over all the initial states “i”.

If the matrix elements do not depend on k,

I (~ω) ≈2π

~|Mfi |

2n(Ef − ~ω) .

The photoemission (and other) spectra thus reflect DOS weightedby transition matrix elements Mfi .

Matrix elements: dipole approximation

Transition matrix element: Mfi ≈⟨

ψf

∣

∣

∣e± i

~q·x ǫ · p

∣

∣

∣ψi

⟩

.

Taylor expansion: e±i

~q·x = 1 ± i

~q · x + 1

2(i~q · x)2 + . . .

If the quantum system is localised within a characteristic length a

(the processes occur at this length scale),

1

~aq ≪ 1 ⇔ a ≪

c

ω⇔ a ≪

λ

2π,

the i~q · x and higher order terms can be neglected.

We thus have just Mfi ≈ ǫ · 〈ψf |p|ψi 〉 .

This is called the dipole approximation.

Other forms of matrix elements

“Primordial” dipole matrix element:

Mfi ≈ ǫ · 〈ψf |p|ψi〉 .

Using the coordinate-momentum commutation relations (and fewother tricks), the matrix element in dipole approximatin can bewritten as

Mfi ≈ ǫ · 〈ψf |p|ψi〉 = imωǫ · 〈ψf |r|ψi 〉 =i

ωǫ · 〈ψf |∇V |ψi 〉

In most situations, all these forms are equivalent.

For practical purposes, the ǫ · 〈ψf |∇V |ψi〉 form is mostly used inPES.

PES and DOS examples: when the nature is nice

PRB 8, 2786 (1973)

Si GaP

Photoemission spectra: Matrix elements effect

Phys. Scr. T109, 61 (2004)

PRB 8, 2786 (1973)

By varying ~ω we can put emphasis on one element or another.

Inverse photoemission

Bremsstrahlung isochromat spectroscopy (BIS).

EF

Electrons arrive from external source,they are decelerated in the vicinity ofions, bremsstrahlung is emitted.

By measuring bremsstrahlung of thesame energy and varying the energyof the incoming electrons, theunoccupied band is scanned.

Transition probability is proportionalto unoccupied DOS.Limited angular momentumselectivity.

BIS: Example

Different bremsstrahlung energyresults in different weights of s-,p- and d -DOS in the spectrum

The 1487 eV isochromat and5415 eV isochromat probe thesame DOS but the matrixelements are different.

PRB 44, 4832 (1991)

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

ARPES — basic principle

Ejected photoelectron with energyEkin is detected.

Wave-vector is determined by theenergy Ekin and the direction θ φ.

Wave vector of the photoelectron invacuum:

Kout,x =1

~

√

2mEkin sin θ cosφ

Kout,y =1

~

√

2mEkin sin θ sinφ

Kout,z =1

~

√

2mEkin cos θ

ARPES — necessity of “non-conservation of momentum”

(Conservation) laws must be obeyed: E(system)f − E

(system)i = ~ω

k(system)f − k

(system)i = kγ

The problem: change of electron momentum cannot be catched upby change of incoming photon momentum (|k|γ = 2π

λ).

Typical crystal momentum: 2π/a ≈ 2A−1

Typical momenta of photons in incoming UV light:

~ω = 100 eV ⇒ |k|γ = 0.05 A−1

~ω = 21.2 eV ⇒ |k|γ = 0.008 A−1

Photon provides the electron with the energy, crystal potentialprovides the momentum needed for reaching the excited state.

For individual electron the crystal momentum is conserved, i.e.,momentum k is conserved up to a reciproval lattice vector G.

Three steps model of photoemission

Not very accurate but transparent and intuitive.

Hufner, Photoelectron spectroscopy (1994)

1. Photoexcitation of anelectron in the bulk.

2. Propagation of the excitedelectron to the surface.

3. Escape of the electron fromthe bulk into the vacuum.

Recovering εnk and k (1)

EF

EV

Eo

Ef

Ekin

Ei

ki

ki+ G

Kout

�

V0

Conservation of energy: Ekin = ~ω − Φ− (EF − Ei )

Recovering εnk and k (2)

EF

EV

Eo

Ef

Ekin

Ei

ki

ki+ G

Kout

�

V0

Momentum:

Parallel component k‖ is conserved up to a reciprocal lattice vector:

k‖ = Kout,‖ − G‖ .

Recovering εnk and k (3)

EF

EV

Eo

Ef

Ekin

Ei

ki

ki+ G

Kout

�

V0

Perpendicular component can be recovered if assumptions aboutthe bulk final state (in the solid) are made.

Assuming the free-electron-like character of the final state, onegets (in the extended zone scheme):

Ef − E0 =~2k2

2m=

~2

2m

[

(k⊥ + G⊥)2 + (k‖ +G‖)

2]

.

Recovering εnk and k (4)

EF

EV

Eo

Ef

Ekin

Ei

ki

ki+ G

Kout

�

V0

Energy balance: Ef − E0 = Ekin + (EV − E0)

Free-electron approx.: Ef − E0 = ~2

2m

[

(k⊥ + G⊥)2 + (k‖ + G‖)

2]

.

Recovering εnk and k (5)

~2

2m

[

(k⊥ + G⊥)2 + (k‖ + G‖)

2]

= Ekin + (EV − E0)

k⊥ + G⊥ =

√

2m

~2[Ekin + (EV − E0)] − K 2

out,‖

=

√

2m

~2

[

Ekin + (EV − E0) − Ekin sin2 θ]

=

√

2m

~2[Ekin cos2 θ + (EV − E0)]

=

√

2m

~2[Ekin cos2 θ + V0]

Few more notes

EF

EV

Eo

Ef

Ekin

Ei

ki

ki+ G

Kout

�

V0

k⊥ + G⊥ =

√

2m

~2[Ekin cos2 θ + V0]

The inner potential V0 has to be determined by an educated quess(say, by fitting it so that experiment matches the theory or byimposing symmetry requirements).

Weak point:Nearly-free electron approximation for the final bulk states willwork well only for “nice” materials (such as alkali or simple metals)and/or for high energies.

Surface sensitivity

To probe bulk, either very low ~ω (threshold PES) or very high ~ω(HAXPES) is needed.

Surface states vers. bulk states

◮ Surface states have no dispertion along k⊥.

◮ Energies and momenta of surface and bulk states cannotoverlap (otherwise, it would be a bulk state. . . )

◮ Surface state have sharper linewidths (DOS in surface layersin more atomic-like).

Example of εnk reconstruction: HgSe(110)

Model of nearly-free electron final states:

k⊥ =√

2m~2 (Ekin + V0)− G⊥

Normal ARPES (θ=0◦).Orlowski et al. 2001

Example of εnk reconstruction: HgTe(110)

Model of nearly-free electron final states:

k⊥ =√

2m~2 (Ekin + V0)− G⊥

Normal ARPES (θ=0◦).Orlowski et al. 2001

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

One step model of photoemission: Less is more

Not so intuitive but moreaccurate.

No hand-waving arguments, onehas to calculate it.

Why:A lot of spectra, a lot of details,a lot of opportunities to find adisagreement (a scientificprediction should be falsifiable).

One step model: Formal description

Probability of a radiative transition:

I (~ω) ≈∣

∣

⟨

εf , k‖ |Hint|ψi

⟩∣

∣

2δ(εf − εi ± ~ω) .

The proper final state is an “inverse LEED state” |εf , k‖〉:

◮ at infinity, it approaches a free-electron with parallelmomentum k‖ and kinetic energy εf ,

◮ in the solid, it attenuates when going deep beneath the surfaceto account for the finite mean-free path of the photoelectron.

The attenuation of the final state is achieved through an imaginarycomponent of the effective one-electron potential.

The |εf , k‖〉 state should be computed accounting for the surfacebarrier. This can be conveniently achieved via multiple-scattering(a.k.a. KKR) or Green function formalism.

One step model: spectrum of Cu(111)

Hufner, Photoelectron spectroscopy (1994)

Good agreement between LDA calculation and theory.

(Dipole) selection rules: where do they come from?

Dipole transition matrix element: ǫ · 〈ψf |r|ψi 〉

Recall and analogy:Integral of a product of an even and an odd functions will beidentically zero.

Likewise:If wave functions |ψi 〉 and |ψf 〉 have certain symmetries, the(dipole) matrix element will be identically zero.

Such transitions are called forbidden transitions.

Transitions forbidden by the dipole selection rule are accessible via

higher-order terms in the exp(± i

~q · x) expansion, their intensity is

usually much less than intensity of the dipole-allowed transitions.

Selection rules in PES: what do they do?

Assuming a sample with a mirror plane,photoemission from a dx2−y2 orbital.

To have non-vanishing intensity, thewhole integrand must be even w.r.t. themirror plane.

◮ Plane wave ψf 〉 is always evenw.r.t. sample mirror plane.

◮ ǫ · r is even if ǫ is in-plane and oddif ǫ is perpendicular to the plane

◮ |ψi 〉 may be in even or odd (here itis even).

Different polarizations probe differentstates.

ǫ · 〈ψf |r|ψi 〉

Example: polarization-dependence of ARPES of Cu(111)

Ludders et al. JPCM 13, 8587

(2001)

Calculated results for the total photocurrent at 21.2 eV photonenergy.

Depth-dependence of spectra: Where the bulk begins?

Ludders et al. JPCM 13, 8587

(2001)

Beyond the order of five layers one basically recovers bulkbehaviour.

Probing surface, probing bulk

By varying the energy, different depths are probed.

~ω=1000 eVFe(001)

8 ML MgO/Fe(001)

~ω=6000 eVFe(001)

8 ML MgO/Fe(001)

C. Felser et al.

Fe-related features recovered for high photon energies ⇒ access toburried interfaces.

Photon momentum effect

For (hard) x-rays, photon momentum |k|γ = 2πλ

cannot beneglected any more.

PRB 77, 045126 (2008)

Outline

Spectroscopy: probe for DOS and more

Probing the electronic structure: ARPES

Calculating (AR)PES: one-step model

What if the effective one-electron approach fails?

One step model: spectrum of Ni

LDA calculation: very poor agreement !

Going beyond the LDA

The LDA is a goose that lays golden eggs.

Direct approaches to many-body physics are not even close to being

materially-specific (save few “singular cases”).

You want to leave as much of LDA it as possible.

LDA+something methods: electron-electron interaction is addedexplicitly on top of the LDA Hamiltonian.

You describe localized electrons in a suitable LDA-generated basisand add Hubbard U and J to describe on-site interaction betweenelectrons.

Caveat: The U and J values are basis-dependent, therefore notblindly transferrable from one calculations to the another.

Double counting problem: Some electron-electron interaction has been

included in the LDA already so care has to be taken not to count it

twice. An educated guess it needed.

Beyond (effective) one-electron description (1)

One electron formalism:

I (~ω) ≈∣

∣

⟨

Ef , k‖ |Hint|ψi

⟩∣

∣

2δ(Ef − Ei − ~ω) .

Assuming N interacting electrons:

I (~ω) ≈∣

∣

∣

⟨

Ψ(N)f |Hint|Ψ

(N)i

⟩∣

∣

∣

2δ(E

(N)f − E

(N)i − ~ω) .

Sudden approximation: Interaction of the excited outgoingphotoelectron with the rest system is neglected.

|Ψ(N)i 〉 = |Ψ

(N)0 〉 ground state

|Ψ(N)f 〉 = a

†f ,k‖

|Ψ(N−1)S 〉 a

†f ,k‖

creates |εf , k‖〉

Beyond (effective) one-electron description (2)

Factorization into single particle and many-body terms:

I (~ω) ≈∑

mm′

∑

S

⟨

φf ,k‖ |Hint|φm

⟩⟨

Ψ(N)0 |a†m|Ψ

(N−1)S

⟩

×δ(εf − ~ω + E(N−1)S − E

(N)0 )

×⟨

Ψ(N−1)S |m′ |Ψ

(N)0

⟩⟨

φm′ |H†int|φf ,k‖

⟩

I (~ω) ≈∑

mm′

⟨

φf ,k‖ |Hint|φm

⟩

Amm′ (εf − ~ω)

⟨

φm′ |H†int|φf ,k‖

⟩

Amm′ (ε) :=

∑

S

⟨

Ψ(N)0

∣

∣

∣a†m

∣

∣

∣Ψ

(N−1)S

⟩

δ(ǫ+ E(N−1)S − E

(N)0 )

×⟨

Ψ(N−1)S

∣

∣am′

∣

∣Ψ(N)0

⟩

After Braun JPCM 16, S2539 (2004)

Spectral function A(ε)

Spectral function A(ε) is a generalization of DOS.

Amm′ (ε) :=

∑

S

⟨

Ψ(N)0

∣

∣

∣a†m

∣

∣

∣Ψ

(N−1)S

⟩

δ(ǫ+ E(N−1)S − E

(N)0 )

×⟨

Ψ(N−1)S

∣

∣am′

∣

∣Ψ(N)0

⟩

m stands for any representation (quantum number). Typically, itdenotes a Bloch state.

Spectral function A(ε) is related to retarded Green function:

Amm′ (ε) := −

1

πImG

(+)

mm′ (ε)

For non-interacting electrons:

Amm(ε) = δ(ε − εm)

Crash course on Green’s functions in quantum theory

To get wave function of a quantum system, we may either solvethe Schrodinger equation

i~∂

∂tΨ = HΨ ,

or “take” the Green’s function (operator) and apply it on the wavefunction at t0,

|Ψ(t)〉 = G (+)(t, t0) |Ψ(t0)〉 .

For time-independent solutions, we can either solve

E |ψ〉 = (H0 + V )|ψ〉 ,

or use a solution of a simpler problem

E |ψ0〉 = H0|ψ0〉 ,

and apply the Green’s function:

|ψ〉 = (1 + G (+)V )|ψ0〉 .

Why Green’s functions?

Working with Green’s functions does not bring anything really new.

Nominally, it is just another formal complication to please themathematical gods.

However, working with Green’s functions is (often) practical inmany-body physics: various approximations can be more easilyintroduced and step-by-step improved in the Green’s functionsformalism.

Spectral function A(ε), important in describing PES, is part of theGreen’s functions world.

Dyson equation

Single-particle situation:Hamiltonian H0 has got a corresponding Green’s function G0,Hamiltonian H0 + V has got Green’s function G :

G (E ) = G0(E ) + G0(E )V G (E ) .

The perturbing potential V does not depend on energy.

Many-body situation:One-particle Green’s function of an interacting many-body systemG (E ) is related to Green’s function of a non-interacting systemG0(E ) via

G (E ) = G0(E ) + G0(E )Σ(E )G (E ) .

Self-energy Σ(E ) depends on energy, it incorporates all themany-body physics.

Self-energy: quick and dirty view

Green’s function of interacting system G (E ), Green’s functin ofnon-interacting system G0(E ), self-energy Σ(E ):

G (E ) = G0(E ) + G0(E )Σ(E )G (E ) .

In representation of single-particle Bloch waves:

G (k, ε) =1

ǫ− εk +Σ(k, ε)

Self-energy Σ(k, ε) describes how binding energies are modified.

Spectral function and spectral lines

I (~ω) ≈∑

mm′

⟨

φf ,k‖ |Hint|φm

⟩

Amm′ (εf − ~ω)

⟨

φm′ |H†int|φf ,k‖

⟩

A(k, ε) =1

π

Σ(k, ε)

[ǫ− εk + ReΣ(k, ε)]2 + [ImΣ(k, ε)]2

ReΣ(k, ε) describes energy shift ofthe spectral peak.

ImΣ(k, ε) describes the change ofthe width of the spectral peak.

Beyond one-electron description: three-step model

States are represented not by single peaks in the spectral functionAk(ε) but by elastic resonances followed by satelite structures.

Describing many electron effects: LDA+DMFT

Self-energy is k-independent, interaction is local.

Bulk Ni: the DOS

ReΣ(ε)

ImΣ(ε)

LDA and LDA+DMFT: angle-integrated spectrum of Ni

Minar et al. PRL 95, 166401 (2005)

Main effects of LDA+DMFT:1. Narrowing of the main peak.

2. Appearence of satellite peak(“lower Hubbard band”).

Fano effect: incoming x-rays arecircularly polarized.

LDA and LDA+DMFT: ARPES of Ni

Braun et al. PRL 97, 227601 (2008)

LDA+DMFT results are not perfect but present a significantimprovement over plain LDA.